Duke Mathematical Journal

Dispersionless Toda and Toeplitz operators

A. Bloch, F. Golse, T. Paul, and A. Uribe

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In this paper we present some results on the dispersionless limit of the Toda lattice equations viewed as the semiclassical limit of an equation involving certain Toeplitz operators. We consider both nonperiodic and periodic boundary conditions. For the nonperiodic case the phase space is the Riemann sphere, while in the periodic case it is the torus $\mathbb {C}/\mathbb {Z}\sp 2$. In both cases we prove precise estimates on the dispersionless limit. In addition, we show that the Toda equations, although they are nonlinear, propagate a Toeplitz operator into an operator arbitrarily close to a Toeplitz operator as long as the Toda partial differential equation (PDE) (dispersionless limit) admits smooth solutions.

Article information

Duke Math. J. Volume 117, Number 1 (2003), 157-196.

First available in Project Euclid: 26 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37K60: Lattice dynamics [See also 37L60]
Secondary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]


Bloch, A.; Golse, F.; Paul, T.; Uribe, A. Dispersionless Toda and Toeplitz operators. Duke Math. J. 117 (2003), no. 1, 157--196. doi:10.1215/S0012-7094-03-11713-5. https://projecteuclid.org/euclid.dmj/1085598341

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